Let a plane P contain two lines r → = i ^ + λ i ^ + j ^ ,   λ ∈ R and r → = - j ^ + μ j ^ - k ^ ,   μ ∈ R . If Q α ,   β ,   γ is the foot of the perpendicular drawn from the point M 1 ,   0 ,   1 to P , then 3 α + β + γ equals .......

Step-by-step solution

Normal vector to the plane containing to given lines n → = i ^ j ^ k ^ 1 1 0 0 1 - 1 = - i ^ + j ^ + k ^ Direction ratios of normal vector = - 1 ,   1 ,   1 . Equation of plane will be - 1 x - 1 + 1 y - 0 + 1 z - 0 = 0 ⇒   x - y - z - 1 = 0 If x ,   y ,   z is foot of perpendicular drawn from M 1 ,   0 ,   1 on the plane, then ⇒ x - 1 1 = y - 0 - 1 = z - 1 - 1 = - 1 - 0 - 1 - 1 3 ⇒ x = 4 3 ,   y = - 1 3 ,   z = 2 3 ⇒ α + β + γ